A new class of one-step nonstandard finite difference methods is developed for first-order ordinary differential equations. The proposed numerical techniques are based on a nonlocal modeling of the right-hand side function and a nonstandard discretization of the time-derivative. This approach leads to significant qualitative improvements in the behavior of the numerical solution. For multi-dimensional autonomous dynamical systems, positive and elementary-stable nonstandard finite difference methods are formulated and analyzed, based on an extension of the nonstandard discretization rules. Applications of the nonstandard finite difference methods to specific biological systems are also presented.